Lecture # 2 - Matrix Algebra

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Lecture # 2 - Matrix Algebra"

Transcription

1 Lecture # - Matrix Algebra Consider our simple macro model Y = C + I + G C = a + by Y C = I + G by + C = a This is an example of system of linear equations and variables In general, a system of x: a 11 x 1 + a 1 x = d 1 a 1 x 1 + a x = d Consider our model of supply and demand Q d = a bp demand equation Q s = c + dp : supply equation Q d = Q s Q d + bp + 0 = a demand equation 0 dp + Q s = c: supply equation Q d + 0 Q s = 0 1

2 And a system of m linear equations and n variables: a 11 x 1 + a 1 x + ::: + a 1n x n = d 1 a 1 x 1 + a x + ::: + a n x n = d :::::::::::::::::::::::::::::::::::::::::::::::::::::: a m1 x 1 + a m x + ::: + a mn x n = d m If there is at least a solution! consistent system If there is NO solution! inconsistent system There are three ingredients in that system of equations Set of coe cients a ij Set of variables x 1 ; x ; :::; x n Set of constant terms d 1 ; d ; :::; d m We can arrange them in three rectangular arrays: A = 6 4 a 11 a 1 ::: a 1n a 1 a ::: a n ::: ::: ::: ::: a m1 a m ::: a mn 7 x = 6 4 x 1 x. x n 7 d = 6 4 d 1 d. d n 7

3 Matrices De nition: A rectangular array of numbers, parameters or variables The number in the i-th row and j-th column (the number in position (i; j)) is denoted by a ij, i.e. A mn = fa ij g If a matrix has m n elements, it is of dimension m n Vector: special matrix. Row vector: Has dimension 1 n Column vector: Has dimension m 1 Other special matrices If m = n : square matrix (DEFINE main diagonal) Identity matrix (I): square matrix with 1 in main diagonal Formally: I has element (I) ii = 1 for all i = 1; :::; n; and (I) ij = 0 for i 6= j Null matrix: matrix N with (N) ij = 0 for all i = 1; :::; m, and j = 1; :::; n. Not necessarily a square matrix. Matrix algebra can be used: 1. To express the system of equations in a compact manner. To nd out whether solution to a system of equations exist. Need n linearly independent equations (where n is the number of elements in x). To obtain the solution if it exists. Note: Matrix linear algebra is applicable only to linear systems of equations

4 Examples: Consider our simple macro model Y = C + I + G C = a + by Y C = I + G by + C = a Matrix of coe cients A 1 1 A = b 1 Consider our model of supply and demand Q d = a bp demand equation Q s = c + dp : supply equation Q d = Q s Q d + bp + 0 = a demand equation 0 dp + Q s = c: supply equation Q d + 0 Q s = 0 Matrix of coe cients A 1 b 0 A = 4 0 d

5 Matrix Operations Matrices are important because they are easy to manipulate Equality: If A mn = fa ij g and B mn = fb ij g are two matrices of same dimension, they are equal if a ij = b ij for all i; j Addition and substraction If A mn = fa ij g and B mn = fb ij g have same dimension, then A + B implies a ij + b ij for all i; j So we add corresponding elements Same with substraction Give example Properties: A + B = B + A A + (B + C) = (A + B) + C Scalar multiplication Multiply each element of matrix A with the constant c Give example

6 Matrix multiplication Multiplication of matrices needs to satisfy a dimensional requirement: If we want A B; we need that the number of columns in A equals the number of rows in B So, if A mn = fa ij g and B pq = fb ij g ; for AB we need n = p If n = q; then the new matrix AB has dimensions m q To continue, consider an example: 1 A = 4 8 B 1 = De ne C = AB : matrix C will have dimension 1 : C = 4 To calculate the element of C; in row i and column j we nd the inner product of row i in A and row j in B Inner product: we take row i in A and row j in B; pair the elements sequentially, multiply each pair, and take the sum of the resulting product Example: to nd c 11 ; we take row 1 in A (1 ) and column 1 in B.( 9) pair each element sequentially: (1 with, and with 9) multiply each pair: (1)()=, ()(9)=7 take the sum of the resulting products: +7==c 11 In the same way, we can nd c 1 = a 1 b 11 + a b 1 = () () + (8) (9) = 8 and c 1 = 0: In general, to nd the value of element c ij we take row i in A and row j in B and n the inner product: c ij = c 11 c 1 c 1 nx a is b sj = a i1 b 1j + a i b j + ::: + a in b nj s=1 6

7 Examples: Consider our simple macro model Y = C + I + G C = a + by Y C = I + G by + C = a In matrix form Ax = d 1 1 A = x = b 1 Y C d = You can verify that Ax gives a matrix I + G a Y C by + C Consider our model of supply and demand Q d = a Q s = Q d = Q s bp demand equation c + dp : supply equation Q d + bp + 0 = a demand equation 0 dp + Q s = c: supply equation Q d + 0 Q s = 0 In matrix form Ax = d 1 b 0 A = 4 0 d 1 x = Q d P Q s d = 4 You can verify that Ax gives a matrix 4 a c 0 Q d + bp dp + Q s Q d Q s 7

8 Properties of multiplication No commutative law: AB 6= BA: Further, some times only one of them is de ned Exceptions B: is a scalar, the identity matrix I, or the inverse of A Associative law (AB)C = A(BC) Left distributive law: A(B + C) = AB + AC Right distributive law: (A + B)C = AC + BC but (A + B)C 6= C(A + B) (ca)b = A(cB) = cab If A mn ; then AI n = A; I m A = A For null matrices: A + N = N + A = A AN = N If A is a square matrix, we canwrite AA = A For an identity matrix of any dimension, II = I = I We cannot write A=B 8

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation

More information

Chapter 7. Matrices. Definition. An m n matrix is an array of numbers set out in m rows and n columns. Examples. ( 1 1 5 2 0 6

Chapter 7. Matrices. Definition. An m n matrix is an array of numbers set out in m rows and n columns. Examples. ( 1 1 5 2 0 6 Chapter 7 Matrices Definition An m n matrix is an array of numbers set out in m rows and n columns Examples (i ( 1 1 5 2 0 6 has 2 rows and 3 columns and so it is a 2 3 matrix (ii 1 0 7 1 2 3 3 1 is a

More information

Math 312 Homework 1 Solutions

Math 312 Homework 1 Solutions Math 31 Homework 1 Solutions Last modified: July 15, 01 This homework is due on Thursday, July 1th, 01 at 1:10pm Please turn it in during class, or in my mailbox in the main math office (next to 4W1) Please

More information

Sample Problems. Practice Problems

Sample Problems. Practice Problems Lecture Notes Factoring by the AC-method page 1 Sample Problems 1. Completely factor each of the following. a) 4a 2 mn 15abm 2 6abmn + 10a 2 m 2 c) 162a + 162b 2ax 4 2bx 4 e) 3a 2 5a 2 b) a 2 x 3 b 2 x

More information

Problem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday.

Problem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday. Math 312, Fall 2012 Jerry L. Kazdan Problem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday. In addition to the problems below, you should also know how to solve

More information

MAT188H1S Lec0101 Burbulla

MAT188H1S Lec0101 Burbulla Winter 206 Linear Transformations A linear transformation T : R m R n is a function that takes vectors in R m to vectors in R n such that and T (u + v) T (u) + T (v) T (k v) k T (v), for all vectors u

More information

Section 5.3. Section 5.3. u m ] l jj. = l jj u j + + l mj u m. v j = [ u 1 u j. l mj

Section 5.3. Section 5.3. u m ] l jj. = l jj u j + + l mj u m. v j = [ u 1 u j. l mj Section 5. l j v j = [ u u j u m ] l jj = l jj u j + + l mj u m. l mj Section 5. 5.. Not orthogonal, the column vectors fail to be perpendicular to each other. 5..2 his matrix is orthogonal. Check that

More information

We shall turn our attention to solving linear systems of equations. Ax = b

We shall turn our attention to solving linear systems of equations. Ax = b 59 Linear Algebra We shall turn our attention to solving linear systems of equations Ax = b where A R m n, x R n, and b R m. We already saw examples of methods that required the solution of a linear system

More information

Orthogonal Projections

Orthogonal Projections Orthogonal Projections and Reflections (with exercises) by D. Klain Version.. Corrections and comments are welcome! Orthogonal Projections Let X,..., X k be a family of linearly independent (column) vectors

More information

Lecture notes on linear algebra

Lecture notes on linear algebra Lecture notes on linear algebra David Lerner Department of Mathematics University of Kansas These are notes of a course given in Fall, 2007 and 2008 to the Honors sections of our elementary linear algebra

More information

Recall that two vectors in are perpendicular or orthogonal provided that their dot

Recall that two vectors in are perpendicular or orthogonal provided that their dot Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal

More information

1 Sets and Set Notation.

1 Sets and Set Notation. LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most

More information

Similar matrices and Jordan form

Similar matrices and Jordan form Similar matrices and Jordan form We ve nearly covered the entire heart of linear algebra once we ve finished singular value decompositions we ll have seen all the most central topics. A T A is positive

More information

9 MATRICES AND TRANSFORMATIONS

9 MATRICES AND TRANSFORMATIONS 9 MATRICES AND TRANSFORMATIONS Chapter 9 Matrices and Transformations Objectives After studying this chapter you should be able to handle matrix (and vector) algebra with confidence, and understand the

More information

GCE Mathematics (6360) Further Pure unit 4 (MFP4) Textbook

GCE Mathematics (6360) Further Pure unit 4 (MFP4) Textbook Version 36 klm GCE Mathematics (636) Further Pure unit 4 (MFP4) Textbook The Assessment and Qualifications Alliance (AQA) is a company limited by guarantee registered in England and Wales 364473 and a

More information

( ) which must be a vector

( ) which must be a vector MATH 37 Linear Transformations from Rn to Rm Dr. Neal, WKU Let T : R n R m be a function which maps vectors from R n to R m. Then T is called a linear transformation if the following two properties are

More information

Eigenvalues, Eigenvectors, Matrix Factoring, and Principal Components

Eigenvalues, Eigenvectors, Matrix Factoring, and Principal Components Eigenvalues, Eigenvectors, Matrix Factoring, and Principal Components The eigenvalues and eigenvectors of a square matrix play a key role in some important operations in statistics. In particular, they

More information

Inner Product Spaces and Orthogonality

Inner Product Spaces and Orthogonality Inner Product Spaces and Orthogonality week 3-4 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,

More information

5 VECTOR GEOMETRY. 5.0 Introduction. Objectives. Activity 1

5 VECTOR GEOMETRY. 5.0 Introduction. Objectives. Activity 1 5 VECTOR GEOMETRY Chapter 5 Vector Geometry Objectives After studying this chapter you should be able to find and use the vector equation of a straight line; be able to find the equation of a plane in

More information

Section 2.4: Equations of Lines and Planes

Section 2.4: Equations of Lines and Planes Section.4: Equations of Lines and Planes An equation of three variable F (x, y, z) 0 is called an equation of a surface S if For instance, (x 1, y 1, z 1 ) S if and only if F (x 1, y 1, z 1 ) 0. x + y

More information

6. Cholesky factorization

6. Cholesky factorization 6. Cholesky factorization EE103 (Fall 2011-12) triangular matrices forward and backward substitution the Cholesky factorization solving Ax = b with A positive definite inverse of a positive definite matrix

More information

CS3220 Lecture Notes: QR factorization and orthogonal transformations

CS3220 Lecture Notes: QR factorization and orthogonal transformations CS3220 Lecture Notes: QR factorization and orthogonal transformations Steve Marschner Cornell University 11 March 2009 In this lecture I ll talk about orthogonal matrices and their properties, discuss

More information

Elements of Abstract Group Theory

Elements of Abstract Group Theory Chapter 2 Elements of Abstract Group Theory Mathematics is a game played according to certain simple rules with meaningless marks on paper. David Hilbert The importance of symmetry in physics, and for

More information

Lecture 2 Linear functions and examples

Lecture 2 Linear functions and examples EE263 Autumn 2007-08 Stephen Boyd Lecture 2 Linear functions and examples linear equations and functions engineering examples interpretations 2 1 Linear equations consider system of linear equations y

More information

1 VECTOR SPACES AND SUBSPACES

1 VECTOR SPACES AND SUBSPACES 1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such

More information

THREE DIMENSIONAL GEOMETRY

THREE DIMENSIONAL GEOMETRY Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,

More information

Math 215 HW #6 Solutions

Math 215 HW #6 Solutions Math 5 HW #6 Solutions Problem 34 Show that x y is orthogonal to x + y if and only if x = y Proof First, suppose x y is orthogonal to x + y Then since x, y = y, x In other words, = x y, x + y = (x y) T

More information

Mathematics Notes for Class 12 chapter 10. Vector Algebra

Mathematics Notes for Class 12 chapter 10. Vector Algebra 1 P a g e Mathematics Notes for Class 12 chapter 10. Vector Algebra A vector has direction and magnitude both but scalar has only magnitude. Magnitude of a vector a is denoted by a or a. It is non-negative

More information

The Singular Value Decomposition in Symmetric (Löwdin) Orthogonalization and Data Compression

The Singular Value Decomposition in Symmetric (Löwdin) Orthogonalization and Data Compression The Singular Value Decomposition in Symmetric (Löwdin) Orthogonalization and Data Compression The SVD is the most generally applicable of the orthogonal-diagonal-orthogonal type matrix decompositions Every

More information

SALEM COMMUNITY COLLEGE Carneys Point, New Jersey 08069 COURSE SYLLABUS COVER SHEET. Action Taken (Please Check One) New Course Initiated

SALEM COMMUNITY COLLEGE Carneys Point, New Jersey 08069 COURSE SYLLABUS COVER SHEET. Action Taken (Please Check One) New Course Initiated SALEM COMMUNITY COLLEGE Carneys Point, New Jersey 08069 COURSE SYLLABUS COVER SHEET Course Title Course Number Department Linear Algebra Mathematics MAT-240 Action Taken (Please Check One) New Course Initiated

More information

3 Orthogonal Vectors and Matrices

3 Orthogonal Vectors and Matrices 3 Orthogonal Vectors and Matrices The linear algebra portion of this course focuses on three matrix factorizations: QR factorization, singular valued decomposition (SVD), and LU factorization The first

More information

expression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method.

expression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method. A polynomial of degree n (in one variable, with real coefficients) is an expression of the form: a n x n + a n 1 x n 1 + a n 2 x n 2 + + a 2 x 2 + a 1 x + a 0 where a n, a n 1, a n 2, a 2, a 1, a 0 are

More information

LINEAR ALGEBRA. September 23, 2010

LINEAR ALGEBRA. September 23, 2010 LINEAR ALGEBRA September 3, 00 Contents 0. LU-decomposition.................................... 0. Inverses and Transposes................................. 0.3 Column Spaces and NullSpaces.............................

More information

Math 115A HW4 Solutions University of California, Los Angeles. 5 2i 6 + 4i. (5 2i)7i (6 + 4i)( 3 + i) = 35i + 14 ( 22 6i) = 36 + 41i.

Math 115A HW4 Solutions University of California, Los Angeles. 5 2i 6 + 4i. (5 2i)7i (6 + 4i)( 3 + i) = 35i + 14 ( 22 6i) = 36 + 41i. Math 5A HW4 Solutions September 5, 202 University of California, Los Angeles Problem 4..3b Calculate the determinant, 5 2i 6 + 4i 3 + i 7i Solution: The textbook s instructions give us, (5 2i)7i (6 + 4i)(

More information

26. Determinants I. 1. Prehistory

26. Determinants I. 1. Prehistory 26. Determinants I 26.1 Prehistory 26.2 Definitions 26.3 Uniqueness and other properties 26.4 Existence Both as a careful review of a more pedestrian viewpoint, and as a transition to a coordinate-independent

More information

a.) Write the line 2x - 4y = 9 into slope intercept form b.) Find the slope of the line parallel to part a

a.) Write the line 2x - 4y = 9 into slope intercept form b.) Find the slope of the line parallel to part a Bellwork a.) Write the line 2x - 4y = 9 into slope intercept form b.) Find the slope of the line parallel to part a c.) Find the slope of the line perpendicular to part b or a May 8 7:30 AM 1 Day 1 I.

More information

Vector Spaces. Chapter 2. 2.1 R 2 through R n

Vector Spaces. Chapter 2. 2.1 R 2 through R n Chapter 2 Vector Spaces One of my favorite dictionaries (the one from Oxford) defines a vector as A quantity having direction as well as magnitude, denoted by a line drawn from its original to its final

More information

RESULTANT AND DISCRIMINANT OF POLYNOMIALS

RESULTANT AND DISCRIMINANT OF POLYNOMIALS RESULTANT AND DISCRIMINANT OF POLYNOMIALS SVANTE JANSON Abstract. This is a collection of classical results about resultants and discriminants for polynomials, compiled mainly for my own use. All results

More information

Linear Algebra Done Wrong. Sergei Treil. Department of Mathematics, Brown University

Linear Algebra Done Wrong. Sergei Treil. Department of Mathematics, Brown University Linear Algebra Done Wrong Sergei Treil Department of Mathematics, Brown University Copyright c Sergei Treil, 2004, 2009, 2011, 2014 Preface The title of the book sounds a bit mysterious. Why should anyone

More information

Using the ac Method to Factor

Using the ac Method to Factor 4.6 Using the ac Method to Factor 4.6 OBJECTIVES 1. Use the ac test to determine factorability 2. Use the results of the ac test 3. Completely factor a trinomial In Sections 4.2 and 4.3 we used the trial-and-error

More information

ISOMETRIES OF R n KEITH CONRAD

ISOMETRIES OF R n KEITH CONRAD ISOMETRIES OF R n KEITH CONRAD 1. Introduction An isometry of R n is a function h: R n R n that preserves the distance between vectors: h(v) h(w) = v w for all v and w in R n, where (x 1,..., x n ) = x

More information

Section 6.1 Factoring Expressions

Section 6.1 Factoring Expressions Section 6.1 Factoring Expressions The first method we will discuss, in solving polynomial equations, is the method of FACTORING. Before we jump into this process, you need to have some concept of what

More information

Factoring Guidelines. Greatest Common Factor Two Terms Three Terms Four Terms. 2008 Shirley Radai

Factoring Guidelines. Greatest Common Factor Two Terms Three Terms Four Terms. 2008 Shirley Radai Factoring Guidelines Greatest Common Factor Two Terms Three Terms Four Terms 008 Shirley Radai Greatest Common Factor 008 Shirley Radai Factoring by Finding the Greatest Common Factor Always check for

More information

Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.

Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. Algebra 2 - Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers - {1,2,3,4,...}

More information

The Characteristic Polynomial

The Characteristic Polynomial Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem

More information

Continuous Groups, Lie Groups, and Lie Algebras

Continuous Groups, Lie Groups, and Lie Algebras Chapter 7 Continuous Groups, Lie Groups, and Lie Algebras Zeno was concerned with three problems... These are the problem of the infinitesimal, the infinite, and continuity... Bertrand Russell The groups

More information

Logistic Regression. Jia Li. Department of Statistics The Pennsylvania State University. Logistic Regression

Logistic Regression. Jia Li. Department of Statistics The Pennsylvania State University. Logistic Regression Logistic Regression Department of Statistics The Pennsylvania State University Email: jiali@stat.psu.edu Logistic Regression Preserve linear classification boundaries. By the Bayes rule: Ĝ(x) = arg max

More information

IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction

IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible

More information

Subspaces of R n LECTURE 7. 1. Subspaces

Subspaces of R n LECTURE 7. 1. Subspaces LECTURE 7 Subspaces of R n Subspaces Definition 7 A subset W of R n is said to be closed under vector addition if for all u, v W, u + v is also in W If rv is in W for all vectors v W and all scalars r

More information

Algebra 2 PreAP. Name Period

Algebra 2 PreAP. Name Period Algebra 2 PreAP Name Period IMPORTANT INSTRUCTIONS FOR STUDENTS!!! We understand that students come to Algebra II with different strengths and needs. For this reason, students have options for completing

More information

LINEAR ALGEBRA W W L CHEN

LINEAR ALGEBRA W W L CHEN LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,

More information

CONTROLLABILITY. Chapter 2. 2.1 Reachable Set and Controllability. Suppose we have a linear system described by the state equation

CONTROLLABILITY. Chapter 2. 2.1 Reachable Set and Controllability. Suppose we have a linear system described by the state equation Chapter 2 CONTROLLABILITY 2 Reachable Set and Controllability Suppose we have a linear system described by the state equation ẋ Ax + Bu (2) x() x Consider the following problem For a given vector x in

More information

3. INNER PRODUCT SPACES

3. INNER PRODUCT SPACES . INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.

More information

Chapter 7 Nonlinear Systems

Chapter 7 Nonlinear Systems Chapter 7 Nonlinear Systems Nonlinear systems in R n : X = B x. x n X = F (t; X) F (t; x ; :::; x n ) B C A ; F (t; X) =. F n (t; x ; :::; x n ) When F (t; X) = F (X) is independent of t; it is an example

More information

1.3 Polynomials and Factoring

1.3 Polynomials and Factoring 1.3 Polynomials and Factoring Polynomials Constant: a number, such as 5 or 27 Variable: a letter or symbol that represents a value. Term: a constant, variable, or the product or a constant and variable.

More information

Factoring Polynomials

Factoring Polynomials UNIT 11 Factoring Polynomials You can use polynomials to describe framing for art. 396 Unit 11 factoring polynomials A polynomial is an expression that has variables that represent numbers. A number can

More information

Section 1.4. Lines, Planes, and Hyperplanes. The Calculus of Functions of Several Variables

Section 1.4. Lines, Planes, and Hyperplanes. The Calculus of Functions of Several Variables The Calculus of Functions of Several Variables Section 1.4 Lines, Planes, Hyperplanes In this section we will add to our basic geometric understing of R n by studying lines planes. If we do this carefully,

More information

1.5. Factorisation. Introduction. Prerequisites. Learning Outcomes. Learning Style

1.5. Factorisation. Introduction. Prerequisites. Learning Outcomes. Learning Style Factorisation 1.5 Introduction In Block 4 we showed the way in which brackets were removed from algebraic expressions. Factorisation, which can be considered as the reverse of this process, is dealt with

More information

Factoring Polynomials: Factoring by Grouping

Factoring Polynomials: Factoring by Grouping OpenStax-CNX module: m21901 1 Factoring Polynomials: Factoring by Grouping Wade Ellis Denny Burzynski This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 3.0

More information

Direct Methods for Solving Linear Systems. Matrix Factorization

Direct Methods for Solving Linear Systems. Matrix Factorization Direct Methods for Solving Linear Systems Matrix Factorization Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University c 2011

More information

College Algebra. Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGraw-Hill, 2008, ISBN: 978-0-07-286738-1

College Algebra. Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGraw-Hill, 2008, ISBN: 978-0-07-286738-1 College Algebra Course Text Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGraw-Hill, 2008, ISBN: 978-0-07-286738-1 Course Description This course provides

More information

Example SECTION 13-1. X-AXIS - the horizontal number line. Y-AXIS - the vertical number line ORIGIN - the point where the x-axis and y-axis cross

Example SECTION 13-1. X-AXIS - the horizontal number line. Y-AXIS - the vertical number line ORIGIN - the point where the x-axis and y-axis cross CHAPTER 13 SECTION 13-1 Geometry and Algebra The Distance Formula COORDINATE PLANE consists of two perpendicular number lines, dividing the plane into four regions called quadrants X-AXIS - the horizontal

More information

7. LU factorization. factor-solve method. LU factorization. solving Ax = b with A nonsingular. the inverse of a nonsingular matrix

7. LU factorization. factor-solve method. LU factorization. solving Ax = b with A nonsingular. the inverse of a nonsingular matrix 7. LU factorization EE103 (Fall 2011-12) factor-solve method LU factorization solving Ax = b with A nonsingular the inverse of a nonsingular matrix LU factorization algorithm effect of rounding error sparse

More information

Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

More information

The Projection Matrix

The Projection Matrix The Projection Matrix David Arnold Fall 996 Abstract In this activity you will use Matlab to project a set of vectors onto a single vector. Prerequisites. Inner product (dot product) and orthogonal vectors.

More information

Recall the basic property of the transpose (for any A): v A t Aw = v w, v, w R n.

Recall the basic property of the transpose (for any A): v A t Aw = v w, v, w R n. ORTHOGONAL MATRICES Informally, an orthogonal n n matrix is the n-dimensional analogue of the rotation matrices R θ in R 2. When does a linear transformation of R 3 (or R n ) deserve to be called a rotation?

More information

Integer Factorization using the Quadratic Sieve

Integer Factorization using the Quadratic Sieve Integer Factorization using the Quadratic Sieve Chad Seibert* Division of Science and Mathematics University of Minnesota, Morris Morris, MN 56567 seib0060@morris.umn.edu March 16, 2011 Abstract We give

More information

Discrete Convolution and the Discrete Fourier Transform

Discrete Convolution and the Discrete Fourier Transform Discrete Convolution and the Discrete Fourier Transform Discrete Convolution First of all we need to introduce what we might call the wraparound convention Because the complex numbers w j e i πj N have

More information

Linear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices

Linear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices MATH 30 Differential Equations Spring 006 Linear algebra and the geometry of quadratic equations Similarity transformations and orthogonal matrices First, some things to recall from linear algebra Two

More information

Online EFFECTIVE AS OF JANUARY 2013

Online EFFECTIVE AS OF JANUARY 2013 2013 A and C Session Start Dates (A-B Quarter Sequence*) 2013 B and D Session Start Dates (B-A Quarter Sequence*) Quarter 5 2012 1205A&C Begins November 5, 2012 1205A Ends December 9, 2012 Session Break

More information

4. FIRST STEPS IN THE THEORY 4.1. A

4. FIRST STEPS IN THE THEORY 4.1. A 4. FIRST STEPS IN THE THEORY 4.1. A Catalogue of All Groups: The Impossible Dream The fundamental problem of group theory is to systematically explore the landscape and to chart what lies out there. We

More information

Section 13.5 Equations of Lines and Planes

Section 13.5 Equations of Lines and Planes Section 13.5 Equations of Lines and Planes Generalizing Linear Equations One of the main aspects of single variable calculus was approximating graphs of functions by lines - specifically, tangent lines.

More information

C relative to O being abc,, respectively, then b a c.

C relative to O being abc,, respectively, then b a c. 2 EP-Program - Strisuksa School - Roi-et Math : Vectors Dr.Wattana Toutip - Department of Mathematics Khon Kaen University 200 :Wattana Toutip wattou@kku.ac.th http://home.kku.ac.th/wattou 2. Vectors A

More information

Factorization Theorems

Factorization Theorems Chapter 7 Factorization Theorems This chapter highlights a few of the many factorization theorems for matrices While some factorization results are relatively direct, others are iterative While some factorization

More information

FUZZY CLUSTERING ANALYSIS OF DATA MINING: APPLICATION TO AN ACCIDENT MINING SYSTEM

FUZZY CLUSTERING ANALYSIS OF DATA MINING: APPLICATION TO AN ACCIDENT MINING SYSTEM International Journal of Innovative Computing, Information and Control ICIC International c 0 ISSN 34-48 Volume 8, Number 8, August 0 pp. 4 FUZZY CLUSTERING ANALYSIS OF DATA MINING: APPLICATION TO AN ACCIDENT

More information

SF2940: Probability theory Lecture 8: Multivariate Normal Distribution

SF2940: Probability theory Lecture 8: Multivariate Normal Distribution SF2940: Probability theory Lecture 8: Multivariate Normal Distribution Timo Koski 24.09.2015 Timo Koski Matematisk statistik 24.09.2015 1 / 1 Learning outcomes Random vectors, mean vector, covariance matrix,

More information

Boolean Algebra Part 1

Boolean Algebra Part 1 Boolean Algebra Part 1 Page 1 Boolean Algebra Objectives Understand Basic Boolean Algebra Relate Boolean Algebra to Logic Networks Prove Laws using Truth Tables Understand and Use First Basic Theorems

More information

Bias in the Estimation of Mean Reversion in Continuous-Time Lévy Processes

Bias in the Estimation of Mean Reversion in Continuous-Time Lévy Processes Bias in the Estimation of Mean Reversion in Continuous-Time Lévy Processes Yong Bao a, Aman Ullah b, Yun Wang c, and Jun Yu d a Purdue University, IN, USA b University of California, Riverside, CA, USA

More information

SOLVING QUADRATIC EQUATIONS - COMPARE THE FACTORING ac METHOD AND THE NEW DIAGONAL SUM METHOD By Nghi H. Nguyen

SOLVING QUADRATIC EQUATIONS - COMPARE THE FACTORING ac METHOD AND THE NEW DIAGONAL SUM METHOD By Nghi H. Nguyen SOLVING QUADRATIC EQUATIONS - COMPARE THE FACTORING ac METHOD AND THE NEW DIAGONAL SUM METHOD By Nghi H. Nguyen A. GENERALITIES. When a given quadratic equation can be factored, there are 2 best methods

More information

AIP Factoring Practice/Help

AIP Factoring Practice/Help The following pages include many problems to practice factoring skills. There are also several activities with examples to help you with factoring if you feel like you are not proficient with it. There

More information

NSM100 Introduction to Algebra Chapter 5 Notes Factoring

NSM100 Introduction to Algebra Chapter 5 Notes Factoring Section 5.1 Greatest Common Factor (GCF) and Factoring by Grouping Greatest Common Factor for a polynomial is the largest monomial that divides (is a factor of) each term of the polynomial. GCF is the

More information

Numerical Analysis Lecture Notes

Numerical Analysis Lecture Notes Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number

More information

(!' ) "' # "*# "!(!' +,

(!' ) ' # *# !(!' +, MATLAB is a numeric computation software for engineering and scientific calculations. The name MATLAB stands for MATRIX LABORATORY. MATLAB is primarily a tool for matrix computations. It was developed

More information

Chapter 2: Systems of Linear Equations and Matrices:

Chapter 2: Systems of Linear Equations and Matrices: At the end of the lesson, you should be able to: Chapter 2: Systems of Linear Equations and Matrices: 2.1: Solutions of Linear Systems by the Echelon Method Define linear systems, unique solution, inconsistent,

More information

Mathematics Review for MS Finance Students

Mathematics Review for MS Finance Students Mathematics Review for MS Finance Students Anthony M. Marino Department of Finance and Business Economics Marshall School of Business Lecture 1: Introductory Material Sets The Real Number System Functions,

More information

Section 1.1. Introduction to R n

Section 1.1. Introduction to R n The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to

More information

3. Let A and B be two n n orthogonal matrices. Then prove that AB and BA are both orthogonal matrices. Prove a similar result for unitary matrices.

3. Let A and B be two n n orthogonal matrices. Then prove that AB and BA are both orthogonal matrices. Prove a similar result for unitary matrices. Exercise 1 1. Let A be an n n orthogonal matrix. Then prove that (a) the rows of A form an orthonormal basis of R n. (b) the columns of A form an orthonormal basis of R n. (c) for any two vectors x,y R

More information

x1 x 2 x 3 y 1 y 2 y 3 x 1 y 2 x 2 y 1 0.

x1 x 2 x 3 y 1 y 2 y 3 x 1 y 2 x 2 y 1 0. Cross product 1 Chapter 7 Cross product We are getting ready to study integration in several variables. Until now we have been doing only differential calculus. One outcome of this study will be our ability

More information

Department of Chemical Engineering ChE-101: Approaches to Chemical Engineering Problem Solving MATLAB Tutorial VI

Department of Chemical Engineering ChE-101: Approaches to Chemical Engineering Problem Solving MATLAB Tutorial VI Department of Chemical Engineering ChE-101: Approaches to Chemical Engineering Problem Solving MATLAB Tutorial VI Solving a System of Linear Algebraic Equations (last updated 5/19/05 by GGB) Objectives:

More information

Solving Simultaneous Equations and Matrices

Solving Simultaneous Equations and Matrices Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. The motivation for considering

More information

Factoring Methods. Example 1: 2x + 2 2 * x + 2 * 1 2(x + 1)

Factoring Methods. Example 1: 2x + 2 2 * x + 2 * 1 2(x + 1) Factoring Methods When you are trying to factor a polynomial, there are three general steps you want to follow: 1. See if there is a Greatest Common Factor 2. See if you can Factor by Grouping 3. See if

More information

Matrix Calculations: Applications of Eigenvalues and Eigenvectors; Inner Products

Matrix Calculations: Applications of Eigenvalues and Eigenvectors; Inner Products Matrix Calculations: Applications of Eigenvalues and Eigenvectors; Inner Products H. Geuvers Institute for Computing and Information Sciences Intelligent Systems Version: spring 2015 H. Geuvers Version:

More information

GROUPS ACTING ON A SET

GROUPS ACTING ON A SET GROUPS ACTING ON A SET MATH 435 SPRING 2012 NOTES FROM FEBRUARY 27TH, 2012 1. Left group actions Definition 1.1. Suppose that G is a group and S is a set. A left (group) action of G on S is a rule for

More information

Algebra 1 If you are okay with that placement then you have no further action to take Algebra 1 Portion of the Math Placement Test

Algebra 1 If you are okay with that placement then you have no further action to take Algebra 1 Portion of the Math Placement Test Dear Parents, Based on the results of the High School Placement Test (HSPT), your child should forecast to take Algebra 1 this fall. If you are okay with that placement then you have no further action

More information

The Greatest Common Factor; Factoring by Grouping

The Greatest Common Factor; Factoring by Grouping 296 CHAPTER 5 Factoring and Applications 5.1 The Greatest Common Factor; Factoring by Grouping OBJECTIVES 1 Find the greatest common factor of a list of terms. 2 Factor out the greatest common factor.

More information

Definitions 1. A factor of integer is an integer that will divide the given integer evenly (with no remainder).

Definitions 1. A factor of integer is an integer that will divide the given integer evenly (with no remainder). Math 50, Chapter 8 (Page 1 of 20) 8.1 Common Factors Definitions 1. A factor of integer is an integer that will divide the given integer evenly (with no remainder). Find all the factors of a. 44 b. 32

More information

1.3 Algebraic Expressions

1.3 Algebraic Expressions 1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,

More information

4.3 Least Squares Approximations

4.3 Least Squares Approximations 18 Chapter. Orthogonality.3 Least Squares Approximations It often happens that Ax D b has no solution. The usual reason is: too many equations. The matrix has more rows than columns. There are more equations

More information

Section 8.8. 1. The given line has equations. x = 3 + t(13 3) = 3 + 10t, y = 2 + t(3 + 2) = 2 + 5t, z = 7 + t( 8 7) = 7 15t.

Section 8.8. 1. The given line has equations. x = 3 + t(13 3) = 3 + 10t, y = 2 + t(3 + 2) = 2 + 5t, z = 7 + t( 8 7) = 7 15t. . The given line has equations Section 8.8 x + t( ) + 0t, y + t( + ) + t, z 7 + t( 8 7) 7 t. The line meets the plane y 0 in the point (x, 0, z), where 0 + t, or t /. The corresponding values for x and

More information

Pythagorean vectors and their companions. Lattice Cubes

Pythagorean vectors and their companions. Lattice Cubes Lattice Cubes Richard Parris Richard Parris (rparris@exeter.edu) received his mathematics degrees from Tufts University (B.A.) and Princeton University (Ph.D.). For more than three decades, he has lived

More information